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The first measurements of dielectron production at midrapidity (|ηc|<0.8) in proton-proton and proton-lead collisions at sNN−−−√ = 5.02 TeV at the LHC are presented. The dielectron cross section is measured with the ALICE detector as a function of the invariant mass mee and the pair transverse momentum pT,ee in the ranges mee < 3.5 GeV/c2 and pT,ee < 8.0 GeV/c2, in both collision systems. In proton-proton collisions, the charm and beauty cross sections are determined at midrapidity from a fit to the data with two different event generators. This complements the existing dielectron measurements performed at s√ = 7 and 13 TeV. The slope of the s√ dependence of the three measurements is described by FONLL calculations. The dielectron cross section measured in proton-lead collisions is in agreement, within the current precision, with the expected dielectron production without any nuclear matter effects for e+e− pairs from open heavy-flavor hadron decays. For the first time at LHC energies, the dielectron production in proton-lead and proton-proton collisions are directly compared at the same sNN−−−√ via the dielectron nuclear modification factor RpPb. The measurements are compared to model calculations including cold nuclear matter effects, or additional sources of dielectrons from thermal radiation.